Matrix method for the Liapunov stability analysis of cyclic discrete mechanical systems

A matrix formalism is developed for the purpose of facilitating the Liapunov stability analysis of discrete, holonomic, mechanical systems with cyclic coordinates and with the Hamiltonian free of explicit time dependence. Matrix expressions are developmed for the kinetic energy, the Routhian, the Hamiltonian, and the quadratic approximation of the dynamic potential energy, with cyclic coordinates, cyclic-coordinate velocities, and cyclic-coordinate generalized momenta not explicitly involved in the last of these functions. The final result is an expression for the quadratic approximation of the dynamic potential energy that is calculated much more readily than by scalar analysis. From the condition for positive-definiteness of this function, Liapunov stability conditions are available. The method is applied to a dual-spin satellite to illustrate the procedure.